Chapter 3.4, Problem 43E

To determine

To find:The instantaneous rate of change of volume of the cube with respect to its side.

Answer to Problem 43E

Correct option that represents instantaneous rate of change of volume of the cube with respect to its side is E that is $3x^2$ .

Explanation of Solution

Given information:

1. Need to find instantaneous rate of change of volume of cube with respect to side of the cube
2. Side of the cube is defined by variable $x$ .
3. four given options of rate change are as follows

1. $x$
2. $3x$
3. $6x$
4. $3x^2$
5. $x^3$

Formula used:

1. Instantaneous rate of change of any function $f\left(x\right)$ is defined by $f^{\prime }\left(x\right)$ using following formula.

  $f^{\prime }\left(x\right)=\underset{h\to 0}{\lim }\frac{f\left(x+h\right)-f(x)}{h}$

2. Volume of cube is defined by function $f\left(x\right)=x^3$ where $x$ is side of the cube.

Calculation:

Function of volume of the cube having side $x$ is as follows

  $f\left(x\right)=x^3$

Instantaneous rate of change of $f\left(x\right)$ is defined by $f^{\prime }\left(x\right)$ .

Evaluate $f^{\prime }\left(x\right)$ .

  $f^{\prime }\left(x\right)=\underset{h\to 0}{\lim }\frac{f\left(x+h\right)-f(x)}{h}$

  $=\underset{h\to 0}{\lim }\frac{{\left(x+h\right)}^3-x^3}{h}$

  $=\underset{h\to 0}{\lim }\frac{x^3+h^3+3x^{2}h+3x{h}^2-x^3}{h}$

  $=\underset{h\to 0}{\lim }\frac{h^3+3x^{2}h+3x{h}^2}{h}$

  $=\underset{h\to 0}{\lim }\frac{h\left(h^2+3x^2+3xh\right)}{h}$

  $=\underset{h\to 0}{\lim }{h}^2+3x^2+3xh$

  $=3x^2$

Hence Instantaneous rate of change of $f\left(x\right)=x^3$ that is volume of cube with respect to $x$ is $3x^2$ that means correct option is D that is $\underset{\_}{3x^2}.$